astronomy

                             The Shapiro delay

Near mass the unit of length shrinks and so distance between fixed points increases.
To bridge the larger distance a photon passing near the sun or a star has to make more oscillations. Think of an oscillation as a step of size one wave length. Taking more steps means that a distant observer sees arrival of the photon delayed.
 
As a particle crossing a gravitational field a photon takes a parabolic route. Red shift occurs when a photon leaves the mass of a star. When nearing mass the opposite happens, which implies gain of energy, reflected in higher frequency. As a wave the photon travels according the principle of "least action", it moves along the shortest optical path. This consists of the smallest number of steps of as large size as possible. Farther away from mass, on the parabole traject, those steps become bigger because the unit of length increases. But the curvature of the parabole should be limited in order to prevent otherwise extra steps. On its own clock of the photon thus a minimum of time is needed.
 
Because a program for proper mathematical representation is not available on the used computer here is mentioned only:
 
    ds exp2 = 1/k exp2 . c exp2 . dt exp2 - k exp2 . dl exp2
 
in which the first and second non-k parts on the right must be as big as possible, respectively as small as possible (for the usual definition of the interval s).
To this suits k = exp1/2 (1 + 2GM/rc exp2).
The second part at the right indicates shrinking of the unit of length and increase of distance.
The first part at the right indicates that the second near mass becomes shorter and thus the total duration of any physical proces there also gets smaller.
 
The second is defined by a frequency of an atom. Higher frequency shows that the transitions of the electrons have more energy and such happens when the atom shrinks, when distances between energy levels become smaller. Because the Coulomb force does not change the electron will move faster in a smaller orbit.
When thus the unit of length in the orbit of an electron shrinks while its speed increases then the second changes square. The dimensions of the propulsion of electromagnetic waves are m/s. So that speed increases near mass.
Pioneers 10 and 11 got less far from the sun than expected with old theory, which is in accordance with the above explanation. 
 
The photon seeks its way where it finds an optimum in circumstances (size of unit of length, size of second, size of speed of light) to harmonize with the principle of least action. The result is well known: a parabole for the distant observer and also for the photon as a particle, but the latter moving as a wave travels straight on in four-dimensional space. Research has to be done on how much the parabole is longer near big mass than a three-dimensional straight line.
 
In old theory it was thought that k equals 1/exp1/2 (1 - 2GM/rc exp2). But then the photon would move hyperbolically: at bigger M and smaller r then c . dt has to increase with as consequence a bigger step nearer to mass and the distant observer would not notice delay on his watch.